Definitely, some items on the top of my list are:
- Random matrix theory --- both asymptotic and non asymptotic; including things like semi-circular law, circular law, and so on. Check out Terry Tao's blog for very nice summaries.
- The resolution of Horn's conjecture (see this nice summary article by R. Bhatia, which also mentions several other nice connections)
- Randomised linear algebra and progress on fast solutions to linear systems (see e.g., the very readable summary in N. Vishnoi's web book)
- Advances in quantum information theory? Though I don't know how much of that I would push into just linear algebra
- Not advances in linear algebra itself, but the gigantic success of basic linear algebra in new areas (machine learning, information retrieval, etc., e.g., Google's PageRank method).